Scope

When translating
sentences in accordance with Russell's theory of descriptions, one needs
sometimes to pay attention to scope. Indeed it is one of the virtues
of the theory that it allows one to make distinctions of scope.

Consider

Buttercup
is not Farmer Giles's cow.

One could translate
this either as:

$x[x is Farmer Giles's cow Ù
¬b=x],

which says that
there is such a thing as Farmer Giles's cow, but that Buttercup is not
it. Or,

¬$x[x is Farmer Giles's cow Ù
b=x],

which is compatible
with there not being any such thing as Farmer Giles's cow.

Actually one might
take the original to mean simply that Buttercup is not a cow belonging
to Farmer Giles; in which case one would not make use of the theory of
descriptions at all.

Consider,

Buttercup
wants to be the first cow to jump over the moon.

We certainly don't
want to translate this as,

$x[x is the first cow to jump over the moon
Ù Buttercup wants to be it].

It is Buttercup
who is deranged, we imagine, and not the speaker. So we should prefer:

Buttercup
wants it to be the case that she is the first cow to jump over the
moon,

Alas, we have not
got the tools to deal with "Buttercup wants it to be the case that";
but we might (perhaps) translate "she is the first cow to jump over
the moon" in accordance with the theory of descriptions.

Again consider:

The
first cow to jump over the moon does not exist,

i.e.

There
is no such thing as the first cow to jump over the moon.

This should not
be translated as,

$x[x is the first cow to jump over the moon
Ù x does not exist],

but rather as,

¬$x x is the first cow to jump over the moon.

Notice,
by the way, that we rendered "exist" by using the existential
quantifier rather than a predicate "x exists". And that is the
normal way of translating "exists".

Thus,

Something
F exists,

Is normally formalised
as,

$xFx.

How about,

Something
exists.

We can formalise
that as:

$xx=x,

which tells us
merely that there is something which is the same thing as itself, which
effectively tells just that something exists.

Notice that it
is not that we cannot treat "exists" as a predicate.
We can perfectly well translate "x exists", as we did just now,
as "x=x". Or (equivalently) we could instead translate it as
"$yx=y" (i.e. "there
is something which x is".)

So we could have
formalised something F exists as:

$x[FxÙx=x]

But that is a long-winded
way of saying something equivalent to "$xFx".

We could indeed
have formalised it as,

$x$y[x=yÙFx].

But that is an
even longer-winded way of saying the same thing.

Notice here in
particular a virtue of Russell's way of treating definite descriptions
compared with translating them with individual constants such as "a".

We have seen that
we can render "There is no such thing as the first cow to jump over
the moon" as,

¬$x x is the first cow to jump over the moon.

And then if, not
trying to be too penetrating, we use "Mx" to mean "x is
a first cow to jump over the moon", and using Russell's theory, we
get,

¬$x[MxÙ"y[My®y=x]],

which has the virtue
of being true.

But if were were
to use "c", say, to mean "the first cow to jump over the
moon", the best we could do would be.

¬$xx=c

or, equivalently,

¬c=c.

But these would
say something false, thanks to the fact that our convention for
using individual contants requires that they name something. (What if
we dropped that convention? Well, that is another story. We would certainly
have to change our tableau rules, since $x¬Fx would no longer follow from ¬Fa.)